Let `A_r ,r=1,2,3, ` , be the points on the number line such that `O A_1,O A_2,O A_3dot` are in `G P ,` where `O` is the origin, and the common ratio of the `G P` be a positive proper fraction. Let `M ,` be the middle point of the line segment `A_r A_(r+1.)` Then the value of `sum_(r=1)^ooO M_r` is equal to
(a) `(O A_1(O S A_1-O A_2))/(2(O A_1+O A_2))` (b) `(O A_1(O A_2+O A_1))/(2(O A_1-O A_2)` (c)`(O A_1)/(2(O A_1-O A_2))` (d) `oo`

Explanation

To solve the problem, we need to analyze the points \( A_r \) on the number line that are in a geometric progression (GP) with the origin \( O \) as a reference point. Let's denote the distance from the origin to the points \( A_1, A_2, A_3 \) as \( OA_1, OA_2, OA_3 \).

### Step 1: Define the Points
Given that \( OA_1, OA_2, OA_3 \) are in GP, we can express them as:
- \( OA_1 = a \)
- \( OA_2 = ar \)
- \( OA_3 = ar^2 \)

where \( a \) is a positive number and \( r \) is the common ratio of the GP, which is a positive proper fraction (i.e., \( 0 < r < 1 \)).

### Step 2: Find the Midpoint \( M_r \)
The midpoint \( M_r \) of the line segment \( A_r A_{r+1} \) can be calculated as:
\[ M_r = \frac{OA_r + OA_{r+1}}{2} \]
Substituting the values we have:
\[ M_r = \frac{OA_r + OA_{r+1}}{2} = \frac{ar^{r-1} + ar^r}{2} = \frac{ar^{r-1}(1 + r)}{2} \]

### Step 3: Calculate the Summation
We need to find the sum:
\[ \sum_{r=1}^{\infty} OM_r = \sum_{r=1}^{\infty} \frac{ar^{r-1}(1 + r)}{2} \]
This can be simplified to:
\[ \sum_{r=1}^{\infty} OM_r = \frac{a(1 + r)}{2} \sum_{r=1}^{\infty} r^{r-1} \]

### Step 4: Evaluate the Infinite Series
The series \( \sum_{r=1}^{\infty} r^{r-1} \) can be recognized as a geometric series. The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio. Here, we have:
\[ S = \frac{a(1 + r)}{2(1 - r)} \]

### Step 5: Final Expression
Combining everything, we find:
\[ \sum_{r=1}^{\infty} OM_r = \frac{a(1 + r)}{2(1 - r)} \]

### Conclusion
After evaluating the expression, we can compare it with the options provided in the question. The correct option that matches our derived expression is option (b):
\[ \frac{OA_1(OA_2 + OA_1)}{2(OA_1 - OA_2)} \]